Villagers will attempt to pick up items even after their inventories are full, all 8 stacks. By "attempt", though, I don't mean walk over to the items and stand near them. I mean the items will act as if they've been picked up, getting attracted to the villager and then disappearing. But, they aren't gone. They stop being rendered and stop counting towards the f3 entity count, but if the player goes where the villager is, they pick the excess potatoes up even though they seemed like they weren't there. The items are just on the floor at the villager's feet, invisible. Even though they were attracted to the villager in the pick-up animation, afterward they move independently of it and will remain if the villager is killed. 

Some possibly useful bits of information:

-If you introduce another villager with inventory space free, they can pick up these invisible items.

-The invisible items can be /killed, it seems.

To sum it up, two things are out of the norm:

-The items no longer being rendered, and

-The fact that the items were drawn to the villager even though it's inventory was full.

What this affects:

I discovered this bug while trying to build a villager breeder where potatoes would be distributed to the villagers by water stream. The first pair of villagers along the stream took all of the potatoes, even when I put 50000 potatoes into the water stream. So, this bug breaks that kind of breeder (to my knowledge the most compact, simplest kind).

That's all the information I have. I hope it helps!

EDIT: I changed the name. Originally, the name of this issue was Infinite Villager Inventory. After some more testing while creating this report I realized that that was not the case, but forgot to change the name upon finishing the report.